January 27, 2015

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1 kigami@i.kyoto-u.ac.jp January 27, 205

2 Contents rank I (3 3 ) II

3 Chapter.... K = R, C { x } K n =. x,, x n K x x =. x n, y = y. y n x n K n x+y K n x+y = αx α K αx K n αx =. αx n = = R C x + y. x n + y n..2. K = R, C U K- (K-vector space) u, v U u + v, α K u U α u (V) (V7) (V) a, b, c U (a + b) + c = a + (b + c) (V2) a, b U a + b = b + a 2

4 (V3) 0 U a U a + 0 = a 0 (zero vector) (V4) a U x U a + x = 0. (V5) a U a = a. K. (V6) a, b U, α K α (a + b) = (α a) + (α b). (V7) a U, α, β K (α + β) a = (α a) + (β a), (αβ) a = α (β a) u, v U u + v u v α K, u U α u u α u αu. U u U 0 u = 0 (0 u) + (0 u) = (0 + 0) u = 0 u..3. K = R, C () R n R-vector space, C n C-vetor space (2) T n (K) = {a 0 + a x + + a n x n a 0, a,, a n K} T n (K) K x n T n (K) K-vector space. (3) C([0, ]) = {f : [0, ] R f } C([0, ]) [0, ] C([0, ]) R-vector space. (4) M m,n (K) = K m n { a a n } M m,n (K) =.. i m, j n a ij K a m a mn.2 R-vector space R 2 e = ( ) x x = R 2 x 2 ( 0 ), e 2 = x = x e + x 2 e 2 ( ) 0 e, e 2 3

5 f = ( ) ( ), f 2 = x = x = ( x x 2 ) = ( x + x 2 2 ) f + ( x x 2 ) f2 2 f, f 2 (e, e 2 ) (f, f 2 ) R 2 (e, e 2, f ) ( ) = e 2 + 2e 2 = e 2 + f ( x.2. ( ). K = R, C U K-vector space e,, e n U () (e,, e n ) (linearly independent) x,, x n K x e + + x n e n = 0 x = = x n = 0 (2) (e,, e n ) U (base, basis) (B), (B2) (B) (e,, e n ) (B2) x U x,, x n K x = x e + + x n e n x (B2). x n K n (e,, e n ) x. (e,, e n ) U x U (e,, e n ) x.,. x n y y n x 2 x U (e,, e n ) x e + + x n e n = y e + + y n e n (x y )e + + (x n y n )e n = 0 (e,, e n ) x y = = x n y n = 0. 4 )

6 .2.2. K = R, C () K-vector space K n 0 0 e = 0., e 2 =.,, e n = (e,, e n ) K n (2) T n (K) (, x, x 2,, x n ) (3) M m,n (K) i m, j n i, j A(i, j) M m,n (K) (i, j)- 0 (A(i, j) i m, j m) M m,n (K). T(R) = x f,, f m T(R) f,, f m N α,, α m R α f + + α m f m T(R) N (f, f m ) T(R).2.3. R-vector space R 3 f, f 2, f 3 f = 0, f 2 =, f 3 = 0 0 (f, f 2, f 3 ) R 3 x. x = x 2 x 3 x = (x x 2 )f + (x 2 x 3 )f 2 + x 3 f 3 k, k 2, k 3 R k f + k 2 f 2 + k 3 f 3 = 0 k + k 2 + k 3 0 k 2 + k 3 k 3 = 0 0 k = k 2 = k 3 = 0 (f, f 2, f 3 ) 5

7 .2.4. g, g 2, g 3 R 3 0 g = 0, g 2 =, g 3 = 0 (g, g 2, g 3 ) R 3. g g 2 g 3 = 0 (g, g 2, g 3 ).. C = {α + βi α, β R} k R, z C kz R-vector space (, i) R-vector space C.2. a, a 2, a 3 R 2 0 a =, a 2 = 2, a 3 = (a, a 2, a 3 ) R (a, a 2, a 3 ) U vetor space, (e,, e n ) U m > n f,, f m U (f,, f m ).3.2. U vector space, (e,, e n ) (f,, f m ) U m = n K = R or C. U K-vector space U U vector space (e,, e n ) U n vector space U K (dimension) n = dim K U K dim K U dim U 6

8 . vetor space U U.2 T (R).3.4. () dim R R n = n. (2) dim C C n = n, dim R C n = 2n. (3) dim K T n (K) = n +. (4) dim K M m,n (K) = mn U vector space e,, e n U 3 () (e,, e n ) (2) π : {,, n} {,, n} (e π(),, e π(n) ) (3) α, β K, α 0, i, j {,, n}, i j e i αe i + βe j , 0, 0, 3 4 R c 3c c 3c c 2 c c+3 2c (4c 3c)/( 34)

9 f = 5 2, f 2 = 2, f 3 = 3 3, f 4 = (f, f 2, f 3, f 4 ) R K = R or C K n n +. n n = a, a 2 K a 2 = 0 0 a + a 2 = 0 a 2 0 a + ( a /a 2 ) a 2 = 0. (a, a 2 ) n K n n a,, a n+ K n i =,, n + a i =. Case i a ni = 0 ã i = a i. a n i a i a ni K n (ã,, ã n+ ) (a,, a n+ ) Case 2 i a ni Step, Step 2, Step 3 Step a n+ a i a n n+ 0 Step 2 a n+ a n n+ a n n+ = Step 3 i =,, n a i a i a ni a n+ a ni = 0 i =,, n 8

10 a a i a n+ a a n+... Step a i a n+.. a n a ni 0 a n n+ a n a n n+ 0 a a a n+ n+ Step 2.. Step 3 ã ã n. a n+ /a nn+ a i a ni a n+ a a n n n+ 0 0 ã i = a i. a n i K n (ã,, ã n ) (a,, a n+ ).3.. K = R or C U K-vector space m = n + f i (e,, e n ) a i a i =. a ni K n U (f,, f n+ ) K n (a,, a n+ ).3.7 (f,, f n+ ).3.8. K = R or C. U K-vector space f,, f m U f,, f m = {u k,, k m K u = k f + + k m f m }.3.9. U vector space f,, f n U (f,, f n ) U (f,, f n ) U = f,, f n U vector space, f,, f m U (f,, f m ) u U u / f,, f m (f,, f m, u) 9

11 . k,, k m, k K k f + + k m f m + ku = 0 u / f,, f m k = 0. (f,, f m ) k = = k m = 0. (f,, f m, u).3.. U vector space, dim U = n f,, f n U (f,, f n ) (f,, f n ) U. u U u / f,, f n (f,, f n, u).3. U = f,, f n..3.9 (f,, f n ).3.2. U vector space, dim U = n m < n f,, f m U (f,, f m ) f m+,, f n U (f,, f n ) U. m < n.3. (f,, f m ) U.3.9 U f,, f m. u U u / f,, f m..3.0 (f,, f m, u) u = f m+ f m+,, f n U (f,, f n ).3. (f,, f n ) U.3.3. k R f (x) = ( k) + 2x + 4x 2, f 2 (x) = 4 + (7 k)x + 0x 2, f 3 (x) = 2 + 4x + (6 + k)x 2 (f, f 2, f 3 ) T 2 (R) k. dim T 2 (R) = 3.3. (f, f 2, f 3 ) (f, f 2, f 3 ) T 2 (R) f, f 2, f 3 (, x, x 2 ) k k k k (k + ) 4 4 2(k + ) 6 + k 2c 2 3c k = 2 0 k 2 (k + ) k 0 2 k c 4 2c 0 0 c 2 2c k 0 2 k 2 0

12 k = 0 k k c+2 c k k 2 k = k 2 3 k 2 (f, f 2, f 3 ) k / {,, 2} K = R or C, U K-vector space V U U (subspace) () u, v V u + v U, (2) k K, u V ku V.4.2. U K-vector space, V U U subspace V U V K-vector space U vector space, a,, a m U a,, a m U subspace a,, a m (a,, a m ) U subspace { x }.4.4. () R 3 V = y x + 3y = z z 0 V R 3 subspace V = 0, 3

13 { x } (2) R 3 V = y x y z V R 3 subspace z V 2 / V. (3) R C C R-vector space R C subspace C C-vector space R C subspace (4)..3-(3) R-vector space C([0, ]) V V = {f f C([0, ]), f(x)dx = 0} V C([0, ]) subspace U vector space, a,, a m U a i 0 i < < i k m (a i,, a ik ) a,, a m a,, a m vector space. B = {(a j,, a jl ) l m, j < < j l m, (a j,..., a jl ) } B a i 0 a i (a i ) B B B l (a i,, a ik ) k l i,, i k i (a i,, a ik, a i ) α,, α k, α α a i + + α k a ik + αa i = 0 (α,, α k, α) (0,, 0, 0). α = 0 (a i, a ik ) (α,, α k ) = (0,, 0). α 0. a i = (α /α)a i + + (α k /α)a ik a i a i,, a ik. a,, a m = a i,, a ik. (a i,, a ik ) a,, a m dim a,, a m = a,, a m m 2

14 .4.6. U vector space, a,, a m U (a,, a m ) (rank), rank (a,, a m ) rank (a,, a m ) m rank (a,, a m ) = dim a,, a m.4.7 (rank ). U K-vector space, a,, a m U () π : {,, m} {,, m} rank (a,, a m ) = rank (a π(),, a π(m) ). (2) α, β K, α 0, i j rank (a,, a i,, a m ) = rank (a,, αa i + βa j,, a m ) a, a 2, a 3, a 4, a 5 R , 2, 4, 2 3, rank (a,, a 5 ) (4c 3c)/8,5c+3 3c c 3c,2c+3c rank (a,, a 5 ) = 3 5c 3 c 2c 3 c,4c c c+8 3c c 3 8c,3c+4 8c 3

15 .4. a, a 2, a 3, a 4 R 4 7, 9 5, , rank (a,, a 4 ) a, a 2, a 3, a 4, a 5 R 4 0, 2, 3 2, 2 3, 2 rank (a,, a 5 ) (a,, a 5 ) a,, a U vector space, V U subspace V dim V dim U.. dim U = n A = {(f,, f m ) f,, f m V, (f,, f m ) }.3. (f,, f m ) A m n. {m (f,, f m ) A} k (a,, a k ) A. a V a / a,, a k.3.0 (a,, a k, a) k V = a,, a k. (a,, a k ) V dim V = k U vector space, V, V 2 U subspaces V + V 2 = {x + x 2 x V, x 2 V 2 } V + V 2 U subspace () V + V 2 V V 2 U subspaces (2) V + V 2 dim V + V 2 = dim V + dim V 2 dim V V 2 4

16 . (2) V V 2 (e,, e k ).3.2 a,, a n V, b,, b m V 2 (e,, e k, a,, a n ), (e,, e k, b,, b m ) V, V 2 V + V = e,, e k, a,, a n, b,, b m. γ e + + γ k e k + α a + + α n a n + β + + β m = 0 x = γ e + + γ k e k + α a + + α n a n = (β b + + β m b m ) x V V 2 α = = α n = β = = β m = 0. x = 0. γ = = γ k = 0. (e,, e k, a,, a n, b,, b m ) V +V 2 dim V + V 2 = k +m+n = (n + k) + (m + k) k = dim V + dim V 2 dim V V U vector space, V, V 2 U subspaces (), (2) () V V 2 = {0} (2) x V + V 2 (x, x 2 ) V V 2 x = x + x 2. V + V 2 (), (2) (3) (3) dim V + V 2 = dim V + dim V 2. () (2): (x, x 2 ), (y, y 2 ) V V 2 x + x 2 = y + y 2 x y = y 2 x 2 V V 2 x = y, x 2 = y 2 (2) (): x V V 2 (x, 0), (0, x) V V 2 (2) (x, 0) = (0, x). x = 0. () (3).4.0-(2) V + V 2 V V 2 V V 2.6. (), (2) V U U subspace () U = R 4 and { x } V = x 2 x 3 x + 2x 2 = 0, x + x 3 + x 4 = 0 x 4 (2) U = T 3 (R) and V = {f f(x)dx = 0} 0 5

17 Chapter K = R or C U, V K-vector space f : U V linear transformation, linear map (L) x, y U f(x + y) = f(x) + f(y), (L2) x U, k K f(kx) = kf(x) K = R or C U, V K-vector space L(U, V ) = U V linear map f, g L(U, V ) f +g : U V x U (f + g)(x) = f(x) + g(x) f + g L(U, V ) k K, f L(U, V ) kf : U V x U (kf)(x) = kf(x) kf L(U, V ) L(U, V ) K-vector space U = V L(U, U) L(U) () f : R 2 R 2 a, b, c, d R ( ( ) x ) ( ) ax + by f = y cx + dy 6

18 f linear map. A = (a ij ) i m, j m M mn (K) f : K n K m ( x ) a x + + a n x n f. =. x n a m x + + a mn x n f linear map 2..4 K n K m linear map L(K n, K m ) M mn (K) (2) R-vector space C ([0, ]) C ([0, ]) = {f f : [0, ] R, f [0, ] } D : C ([0, ]) C ([0, ]) Df = f f f D linear map. (3) f : C C f(z) = z z z f C R-vector space linear map C-vector space linear map U, V K-vector space, f : U V linear map (a,, a n ), (b,, b m ) U, V f(a j ) (b,, b m ) a j. a mj A M mn (K) A = (a ij ) i m, j n x U (a,, a n ) x. x n, f(x) V (b,, b m ) y y m x. = A. x n y. y m A (a,, a n ), (b,, b m ) f (a,, a n ), (b,, b m ) f (U = V (a,, a n ) = (b,, b m ) A (a,, a n ) f 7

19 . x = x a + + x n a n f(x) = x f(a ) + + x n f(a n ). f(a j ) = a j b + + a mj b m f(x) = m ( ( n ) ) a ij x j bi i= j= y i = n j= a ijx j p(x) T 3 (R) F (p)(x) T 2 (R) F (p)(x) = p (x ) p (x) p(x) x F : T 3 (R) T 2 (R) F linear map F (, x, x 2, x 3 ), (, x, x 2 ). linear map F () = 0, F (x) =, F (x 2 ) = 2(x ) = 2x, F (x 3 ) = 3(x ) 2 = 3x 2 6x+3 F (, x, x 2, x 3 ), (, x, x 2 ) p(x) T 3 (R) (F (p))(x) T 3 (R) (F (p))(x) = (x + )p (x) p p F : T 3 (R) T 3 (R) linear map F (, x, x 2, x 3 ) 2.2. f L(R 3 ) f( 0 ) = 0, f( 0 ) =, f( ) 3 = e = 0, e 2 =, e 3 = 0 f (e, e 2, e 3 ) 0 0 8

20 2..6. U, V, W K-vector spaces, f L(U, V ), g L(V, W ) g f : U W linear map U, V, W (a,, a n ), (b,, b m ), (c,, c l ) U, V, W a a n A =.. a m a mn M mn (K), B = b b m.. b l b lm M lm (K) A (a,, a n ), (b,, b m ) f B (b,, b m ), (c,, c l ) g g f (a,, a n ), (c,, c l ) C = (c ij ) i l, j n M ln (K) c ij = m k= b ika kj C = BA B A. g(f(a j )) = g(a j b + + a mj b m ) = a j g(b ) + + a mj g(b m ) m l l m = (a kj b ik c i ) = ( b ik a kj )c i k= i= i= k= 2.3. ( ) ( ) (), (2) 4 5 6, (3) , A A 2, A 3, A =

21 2..7. U, V K-vector space, f L(U, V ) Imf = {f(x) x U}, ker f = {x x U, f(x) = 0} Imf, ker f V, U subspace Imf f (image) ker f f (kernel) f ker f = {0}.. f(x )+f(x 2 ) = f(x +x 2 ), kf(x) = f(kx) Imf, ker f subspaces f f(x) = 0 f(x) = f(0) x = 0. ker f = {0}. ker f = {0} f(x ) = f(x 2 ) f(x x 2 ) = 0. x x 2 ker f x = x 2. f U, V vector spaces, f L(U, V ) dim Imf + dim ker f = dim U. ker f (a,, a m ).3.2 b,, b n U (a,, a m, b,, b n ) U x U x = α a + +α m a m +β b + +β n b n f(x) = β f(b ) + + β m f(b n ) Imf = f(b ),, f(b n ). (f(b ),, f(b n )) β f(b ) + + β n f(b n ) = 0 f(β b + + β n b n ) = 0. β b + + β n b n ker f β = = β n = 0. (f(b ),, f(b n )) Imf dim U = m + n = dim kerf + dim Imf. c,, c k V (f(b ),, f(b n ), c,, c k ) V U (b,, b n, a,, a m ), V (f(b ),, f(b n ), c,, c k ) f ( ) In 0 nm 0 kn 0 km I n M nn (K) n n i, j { i = j δ ij = 0 i j I n = (δ ij ) i,j n. O ij M ij (K) i j 20

22 2..9. U, V vextor spaces, f L(U, V ) n = dim Imf f ( ) In U, V 2.5. U vector space, f L(U) f f = f V = {x x U, f(x) = x} U = ker f V 2.6. U vector space, f L(U) n =, 2, f n L(U) f = f, f n+ = f f n () m Imf m Imf m+ n Imf n = Imf n+ m n Imf n = Imf m (2) m ker f m ker f m+ n ker f n = ker f n+ m n ker f n = ker f m 2.7. M 3 (R) A M 3 (R) t A A f : M 3 (R) M 3 (R) f(a) = (A + t A)/2 f {A A M 3 (R), f(a) = A}, f {f(a) A M 3 (R)} f {A A M 3 (R), f(a) = 0} U, V vector spaces, f L(U, V ) f or g L(U, V ) f g = I V, g f = I U I V, I U V, U g f (inverse) g = f U, V vector spaces, f L(U, V ) () f f (2) f f 2

23 (3) f e,, e m U (e,, e m ) (f(e ),, f(e m )) U V dim U = dim V () ker f = {0} Imf = V. () x, x 2 U f(x ) = f(x 2 ) x = g(f(x )) = g(f(x 2 )) = x 2. f y V f(g(y)) = y y Imf. Imf = V. y V g(x) = y x U x = g(y) g : V U f g = I V, g f = I U. g L(U, V ) y, y 2 V f(g(y ) + g(y 2 )) = f(g(y )) + f(g(y 2 )) = y + y 2. g g f = I V y V, k K g(y ) + g(y 2 ) = g(y + y 2 ). f(kg(y)) = kf(g(y)) = ky g g f = I V kg(y) = g(ky) g L(V, U) f (2) () g (3) k,, k m K U, V K-vector spaces k f(e ) + + k m f(e m ) = 0 g(k f(e ) + + k m f(e m )) = k e + + k m e m = 0. (e,, e m ) k = = k m = 0. (f(e ),, f(e m )) U (e,, e m ) U (f(e ),, f(e m )) V = f(u) = f(e ),, f(e m ). (f(e ),, f(e m )) V U, V vector spaces dim U = dim V f L(U, V ) 4 22

24 () f (2) m N, e,, e m U (e,, e m ) (f(e ),, f(e m )) (3) ker f = {0}. f (4) Imf = V. f. () (2) (3) (2) (4): (e,, e n ) U dim U = dim V = n..3. (f(e ),, f(e n )) V Imf = V. (3) (4): 2..8 dim kerf = 0 dim Imf = dim U = dim V. (3) (): (3) (4) (3) (4) f 2.8. U, V vector spaces f L(U, V ), g L(V, U) g f = I U V = Imf ker g A M mn (K) B M nm (K) AB = I m, BA = I n I k M kk (K) k k A invertible B A (inverse) A A B B, B 2 A AB = AB 2 = I m B = B AB = B AB 2 = B U, V vector spaces (a,, a n ), (b,, b m ) U, V f L(U, V ) f (a,, a n ), (b,, b m ) A M mn (K) f A f (b,, b m ), (a,, a n ) A. f f (b,, b m ), (a,, a n ) B 2..6 B A A A (b,, b m ), (a,, a n ) g L(V, U) g = f (3) A M mn (K) A m = n. 23

25 2.3 rank () U, V f L(U, V ) f (rank), rank f rank f = dim Imf (2) A = (a ij ) M mn (K) i =,, n a j A j a j = a j. a mj rank A rank A = rank (a,, a n ) A (rank), U, V vector spaces, (e,, e n ), (g,, g m ) U, V f L(U, V ) f (e,, e n ), (g,, g m ) A M mn (K) rank f = dim f(e ),, f(e n ) = rank (f(e ),, f(e n )) = rank A () U, V vector spaces, f L(U, V ) f dim U = dim V = rank f (2) A M nn (K) A rank A = n. U, V vector spaces, (e,, e n ), (g,, g m ) U, V f L(U, V ) f (e,, e n ), (g,, g m ) A M mn (K) A i â i = (a i a in ) A =. â â m α, β K α 0 k, l {,..., m} k l (h,, h m ) i / {k, l} h i = g i, h k = g k β α g l, h l = α g l () (h,, h m ) V (2) f (e,, e n ), (h,, h m ) B M mn (K) B i ˆb i = (b i... b in ) { â i i l, ˆbi = (2.) αâ l + βâ k i = l. 24

26 . () g l = αh l g k = h k + βh l (2.2) g,..., g m = h,..., h m. V = h,..., h m. dim V = m (h,..., h m ) V (2) f(e j ) = m i= a ijg j (2.2) f(e j ) = m i=,i k,l a ij h i + αa lj h l + a kj (h k + βh l ) = i=,i l a ij h i + (αa lj + βa kj )h l A, B f rank A = rank f = rank B A (2.3) rank A A rank A M mn (K), A = (a ij ) i m, j n A j- a j = a j. a mj K m A = (a a n ) A i- â i = (a i a in ) A =. â â m a a n â A =.. = (a a n ) =. a m a mn â m rank A = rank (a,, a m ) (2) (rank ). A = (a ij ) i m, j n M mn (K) 25

27 (C) π : {,, m} {,, m} rank A = rank (a π() a π(m) ). (C2) α, β K, α 0, i j rank A = rank (a a i a m ) = rank (a αa i + βa j a m ). (R) σ : {,, n} {,, n} rank A = rank (R2) α, β K, α 0, i j â â σ(). â σ(n).. rank A = rank â i = rank αâ i + βâ j.. â n â n (C), (C2) (Colum) (R), (R2) (Row) (C), (C2), (R), (R2) rank rank â 26

28 c 2 3c c c c 7 c (2c 4 3c)/( 0) r 3 2r r 2r rank 3. 4r 2r/4 2r/4,3r 4r+r 2r+r,3r r rank () (2) 3 2 (3) (4) (5) a a (6) 4 a 0 (7) a (8) a b c d 0 0 a a a a 2 b 2 c 2 d 2 a 3 b 3 c 3 d A M mn (K) k = rank A A (C), (C2), (R), (R2) ( ) Ik O k n k O m k k O m k n k 27

29 B M mn (K) B 0 mn B (C), (C2), (R), (R2) c c n (2.3)... 0 c m c m n. b b j b n b i b ij b in r ir b i b ij b in b b j b n b m b mj b mn b m b mj b mn b ij b i b in d 2 d n... c/b ij d 2 d 22 d 2n b j b b n d m d m2 d mn b mj b m b mn d 2 ir d i r i m... d m 0 c jc jc d j c 2 j n m+n m+n = 2 (m, n) = (, ), (0, 2), (2, 0). m + n = N m + n = N (C), (C2), (R), (R2) A M mn (K) (2.3) C = (c ij ) C = M m n (K) C (C), (C2), (R), (R2) 28

30 2.0. () U, V vector spaces, h L(U), f L(U, V ), h 2 L(V ) h, h 2 rank f = rank h 2 f h (2) A M mn (K), B M nn (K), B 2 M mm (K) B, B 2 rank A = rank B 2 AB 2.. U, V, W f L(U, V ), g L(V, W ) (), (2) () rank g f = rank g V = ker g + Imf (2) rank g f = rank f ker g Imf = {0} 2.2. A M nn (K) A 3 = A A 2 = A 4 rank A = rank A A M nn (K) A n 0, A n = 0 rank A = n 2.4. A M nn (K) rank A + rank (I n A) n A = (a ij ) i m, j n M mn (K) A t A M nm (K) A a a 2 a m t a 2 a 22 a m2 A =... a n a 2n a mn A i â i t A i t â i A j a j t A j t a j â A = (a a n ) =. â m t a t A =. t a n = ( t â tâ m ) 29

31 A M mn (K) rank A = rank t A 2.5. A M mn (K), B M lm (K) t (BA) = t A t B 2.6. (2) A M nn (K) A t A A ( t A) = t (A ) 2.7. K = R C U K- f : U U I U : U U u U I U (u) = u () α K V (α) = {u u U, f(u) = αu} V (α) U (2) α, α 2 K α α 2 V (α ) V (α 2 ) = {0} (3) α, α 2 K α α 2 rank (f α I U ) + rank (f α 2 I U ) dim U K = R or C U K-vector space U = L(U, K) U U (dual space) U = L(U, K) U K-vector space U vector space (a,, a n ) U a i U a i (α a + + α n a n ) = α i (a,, a n ) U U vector space dim U = dim U. (a,, a n ) (a,, a n ) 30

32 . h U h(α a + + α n a n ) = n α i h(a i ) = i= n h(a i )a i (α a + α n a n ) i= h = n i= h(a i)a i. U = a,, a n. (a,, a n ) p = β a + + β n a n p(a i ) = β i p = 0 i β i = 0. (a,, a n ) U, V vector spaces, f L(U, V ) t f : V U t f(h) = h f f : U V, h : V K t f(h) : U K) t f f t f L(V, U ) 2.8. U, V vector spaces f L(U, V ) t f L(V, U ) L(U, V ) L(V, U ) linear U, V vector space (a,, a n ), (b,, b m ) U, V f L(U, V ) f (a,, a n ), (b,, b m ) A f t f L(V, U ) (b,, b m ), (a,, a n ) t A. t f (b,, b m ), (a,, a n ) B = (b ij ) i n, j m t f(b j ) = n i= b ija i. t f(b j )(a k ) = b kj. t f(b j )(a k ) = b j (f(a k )) = b j ( m l= a lkb l ) = a jk. b kj = a jk U, V vector space, f L(U, V ) dim Imf = dim Im t f k = dim Imf ( U (a),, a n ), V Ik 0 (b,, b m ) f k n k 0 m k ( k 0 m k n k ) t Ik 0 f k m k 0 n k k 0 n k m k dim Im t f = k U, V, W vector spaces, f L(U, V ), g L(V, W ) t (g f) = t f t g 3

33 2.20. U, V vector spaces, f L(U, V ) f t f f ( t f) = t (f ) 32

34 Chapter 3 3. I (3 3 ) ( ) a b 2 2- c d ( ) a b ad bc 0 c d a a 2 a A = a 2 a 22 a 23 a 3 a 32 a 33 A rank A = e a b ad bc 0. f c d rank e a b = rank 0 a b f c d 0 c d ( 0 0 0, a, b ( ( ) ( ) ( ) a b a b, ) c d c d 0 c d ad bc 0. 33

35 a 0 a a 2 a 3 a 2 a 3 A = a 2 a 22 a 23 c/a a 2 /a a 22 a 23 a 3 a 32 a 33 a 3 /a a 32 a c a 2 c a 2 /a a 22 a 2 a 2 /a a 23 a 3 a 2 /a 3c a 3 c a 3 /a a 32 a 2 a 3 /a a 33 a 3 a 3 /a 3.. A 0 ( a 22 a 2a 2 )( a33 a 3a 3 ) ( a23 a 3a 2 )( a32 a 2a 3 ) = a a a a a a 22 a 33 a 22 a 3 a 3 a 2 a 2 a 33 a 23 a a 32 + a 23 a 2 a 3 + a 2 a 32 a 3 a A = (a ij ) i,j 3 A (determinant), det A A A = a a 22 a 33 + a 2 a 23 a 3 + a 3 a 2 a 32 a 3 a 22 a 3 a 2 a 2 a 33 a a 23 a A = (a ij ) i,j 3 A = 0.. a 0 a = 0 A 0 a ij 0 (C), (R) a 0 a a 2 a 3 a a 2 a 3 A = a 2 a 22 a 23 a = a 2, a 2 = a 22, a 3 = a 23 a 3 a 32 a 33 a 3 a 32 a 33 A = (a a 2 a 3 ). A = D(a, a 2, a 3 ) () a, b K 3, α, β K D(αa + βb, a 2, a 3 ) = αd(a, a 2, a 3 ) + βd(b, a 2, a 3 ) D(a, αa + βb, a 3 ) = αd(a, a, a 3 ) + βd(a, b, a 3 ) D(a, a 2, αa + βb) = αd(a, a 2, a) + βd(a, a 2, b) 34

36 (2) D(a, a 3, a 2 ) = D(a 2, a, a 3 ) = D(a 3, a 2, a ) = D(a, a 2, a 3 ) (3) α, β K (4) A = t A D(αa + βa 2, a 2, a 3 ) = D(αa + βa 3, a 2, a 3 ) =D(a, αa 2 + βa, a 3 ) = D(a, αa 2 + βa 3, a 3 ) =D(a, a 2, αa 3 + βa ) = D(a, a 2, αa 3 + βa 2 ) =αd(a, a 2, a 3 ). (), (2), (4) (3) () D(αa + βa 2, a 2, a 3 ) = αd(a, a 2, a 3 ) + βd(a 2, a 2, a 3 ) (a 2, a 2, a 3 ) (a 2 a 2 a 3 ) D(a 2, a 2, a 3 ) = 0. (C), (C2), (R), (R2) k k k. k k 4 2c 2 3c k 0 2 = 2 (k + ) k 4 2(k + ) 6 + k = k 0 2 (k + ) 2 4 c 2 2c 0 2 = (k + )( k) k k 0 2 (k + )(k )(k 2) 0 4 = (k + )(k )(k 2) r 2 2r = 35

37 a b b () (2) b a b b b a a + x a + y a + z (4) b + x b + y b + z (5) c + x c + y c + z (b + c) 2 c 2 b 2 (6) c 2 (c + a) 2 a 2 b 2 a 2 (a + b) 2 x x 2 (3) x x 2 x 2 x b + c a a b c + a b c c a + b 3.2. a + b b + c c + a () b + c c + a a + b c + a a + b b + c = 2 a b c b c a c a b (2) b + c c + a a + b = (a b)(b c)(c a) bc ca ab (a + b) 2 ca bc (3) ca (b + c) 2 ab = 2abc(a + b + c)3 bc ab (c + a) 2 b + c a c a b (4) b c c + a b a c b c a a + b = 8abc 3.3. ABC 3 A, B, C cos C cos B cos C cos A cos B cos A = II K = R or C K n n a,, a n K n D n (a,, a n ) K D n : K n K n K (DT), (DT2), (DT3) 36

38 (DT) i {,, n}, α, β K n, α, β K D n (a,, αa + βb,, a n ) i = αd n (a,, a i,, a n ) + βd n (a,, b i,, a n ) (DT2) i < j n D n (a,, a j,, a i,, a n ) i j = D n (a,, a i,, a j,, a n ) i j 0. 0 (DT3) D n (e,, e n ) = e i = i (i = {,, n} n n A = (a a n ) j {,, n} a j A j ) A (determinant) A det A A = D n (a,, a n ) (DT), (DT2), (DT3) D n n n =, 2, 3 D, D2, D3 Dn (DT), (DT2), (DT3) j =,, n a j a 2j a nj a 2j a j =., b j =. a nj D n(a,, a n ) = n ( ) j a j D n (b,, b j,, b n ) j= 37

39 b j b j ( b, b 2,, b n ) = (b 2,, b n ), (b, b 2, b 3,, b n ) = (b, b 3,, b n ). (DT) a,, a n K n, a i = αa + βb a = u v u 2., b = v 2 u 2 v 2., a =., b =. Dn (DT) u n v n u n Dn(a,, a n ) = α ( ) j a j Dn (b,, b j,, a + β j: j n,j i j: j n,j i v n i ( ) j a j D n (b,, b j,, b + ( ) i (αu i + βv i )D n (b,, b i,, b n ) i,, b n ),, b n )) = αdn(a,, a,, a n ) + βdn(a,, b,, a n ) i i (DT2) < i Dn(a i,, a,, a n ) = a i Dn (b 2,, b i, b, b i+,, b n ) i + ( ) i a Dn (b i, b 2,, b i, b i+,, b n ) + ( ) j Dn (b i, b 2,, b j,, b,, b n ) j: j n,j,i (3.) D n (DT2) Dn (b 2,, b i, b, b i+,, b n ) = ( )Dn (b 2,, b, b i, b i+,, b n ) = = ( ) i 2 Dn (b,, b i,, b n ) Dn (b i, b 2,, b i, b i+,, b n ) = ( ) i 2 Dn (b 2,, b n ) Dn (b i, b 2,, b j,, b,, b n ) = ( )Dn (b,, b j,, b n ) 38

40 (3.) D n(a i, a 2,, a,, a n ) = ( )Dn(a,, a n ). i (DT3) Dn Dn (DT), (DT2), (DT3) D n (DT), (DT2), (DT3) () i j a i = a j D n (a,, a n ) = 0. (2) α, β K, i j D n (a,, αa i + βa j,, a n ) = αd n (a,, a i,, a n ) i i (3) π : {,, n} {,, n} sgn π = D n(e π(),, e π(n) ) D n (a π(),, a π(n) ) = sgn πd n (a,, a n ). () a = a 2 (DT2) D n (a, a 2,, a n ) = D n (a 2, a,, a n ). a = a 2 D n (a, a,, a n ) = D n (a, a 2,, a n ). D n (a,, a n ) = 0. (2) (DT) D n (a,, αa i + βa j,, a n ) = αd n (a,, a i,, a n ) + βd n (a,, a j,, a n ). () D n (a,, a j,, a n ) = 0 (2) (3) S n = {π π : {,, n} {,, n}, π } S n n (permutation group) ( π S n ) n (permutation) π S n π = 2 n π() π(2) π(n) i, j n, i j σ i,j S n j k = i, σ i,j (k) = i k = j, k k i, j. σ i,j i, j (i, j) 39

41 π S n (, 2,, n) (π(), π(2),, π(n)) i, j σ i,j i j ( ) () π = (, 2, 3, 4) (,4) (4, 2, 3, ) (,3) (4, 2,, 3) (,2) (4,, 2, 3) ( π = σ,2 ) σ,3 σ,4 2 3 (2) π = 3 2 (, 2, 3) (,3) (3, 2, ) (,2) (3,, 2) (, 2, 3) (,2) (2,, 3) (,3) (2, 3, ) (2,3) (3, 2, ) (,2) (3,, 2) π S n (i, j ),, (i m, j m ) π = σ im,j m σ i,j. sgn π = ( ) m. n n = S = { } 0 ( ) n n π S n π(n) = n π() π(n ) S n π π(n) n σ = σ π(n),n π σ(n) = n. σ π = σ π(n),n σ π sgn π = ( ) m (DT2), (DT3) π S n, (i, j ),, (i m, j m ) (k, l ),, (k p, l p ) π = σ im,jm σ i,j = σ kp,lp σ k,l m, p m p 40

42 . sgn π = ( ) m = ( ) p (3) A = (a ij ), B = (b ij ) M nn (K) a j b j A, B j a j =. a j a nj c j C = BA C j c j =., b j = c nj b j. b nj. D n (c,, c n ) = π S n sgn πa π() a π(2) 2 a π(n) n D n (b,, b n ).. c j = a j b + a 2j b a nj b n n D n (c,, c n ) = D n ( a i b i, c 2,, c n ) n = a i D n (b i, = i = n i,,i n= i = n a i2 2b i2,, c n ) = i 2 = i,i 2 = a i a i2 2 a in nd n (b i, b i2,, b in ) n a i a i2 2D n (b i, b i2, c 3,, c n ) (), (3) D n (b i,, b in ) = ( ) ( ) n n sgn D n (b,, b n ) i i n i i n 0 k l i k = i l (DT), (DT2), (DT3) D n D n (a,, a n ) = sgn πa π() a π(2) 2 a π(n) n π S n 4

43 B = I n D n A, B M nn (K) AB = A B A M nn (K) A A = 0. A A = A.. : A AA = I n. = I n = AA = A A. A 0. : A A = (a a n ), a,, a n K n (a,, a n ) j a j = α a + + α j a j + α j+ a j+ + + a n () A = D n (a,, a j,, a n ) = i=,,n,i j α i D n (a,, a i,, a n ) = 0 j A = (a ij ) M nn (K) A = t A. A = σ S n sgn σa σ() a 2 σ(2) a n σ(n).. t A = (b ij ) b ij = a ji t A = σ S n sgn σa σ() a 2 σ(2) a n σ(n) σ {,, n} σ σ = σ im,j m σ i,j σ = σ i,j σ im,j m sgn σ = sgn σ. π = σ a σ() a 2 σ(2) a n σ(n) = a π() a π(2) 2 a π(n) n 42

44 σ σ S n t A = sgn σa σ() a 2 σ(2) a n σ(n) σ S n = sgn σ a σ () a σ (2) 2 a σ (n) n σ S n = sgn πa π() a π(2) 2 a π(n) n = A π S n ( ) π = S ( ) ( ) π = = (DT), (DT2) A = (a ij ) M nn (K) i â i = (a i a in ) ( t a i K n ) â A =. â n (), (2), (3) () α, β K, t a, t b K n â k = αa + βb â â.. A = α ạ k + β ḅ k.. (2) π S n â n â π(). â π(n) â n â = sgn π. â n 43

45 (3) i j, α, β K â â.. αâ i + βâ j i = α â i i.. â n â n x 0 x 0 x x x x 0 = 0 x x 0 x. x 0 x 0 x 0 x x c+(2c+3c+4c) x x 0 = (2x + ) x x 2c c x x 0 = 4c x c x 0 x 0 x (2x + ) x x x ir r(i=2,3,4) x x x = (4x 2 ) 0 x x 2c+3c 0 x x c x 2c = 4c+x 3c (4x2 ) 0 0 2r 3r = (4x 2 ) (4x 2 )D 4 (e, e 3, e 4, e 2 ) = (4x 2 ). x = /2, / x y () (2) x y y x y x (3) a b c d a b c d a b c d a b c d 44

46 3.5. x x 3 x 2 x x x 3 x 2 x 2 x x 3 = 0 x 3 x 2 x A = (a ij ) i,j n M nn (K) n a 2 a 2 j a 2 j+ a 2n A = ( ) j a j.... j= a n a n j a n j+ a nn (3.2) (i, i ), (i, i 2),, (2, ) a i a in a a n. A = ( ) i a i a i n a i+ a i+ n. a n a nn (3.2) A = ( ) i n ( ) j a ij A ij, (3.3) j= A ij A i j (n ) (n ) a a j a j+ a n.... a A ij = i a i j a i j+ a i n a i+ a i+ j a i+ j+ a i+ n.... a n a n j a n j+ a nn 45

47 3.3.. A = (a ij ) i,j n M nn (K) ã ij = ( ) i+j A ij A (i, j)- (cofactor) A = (a ij ) i,j n M nn (K) { n A (k = l) a kj ã lj = 0 (k l) j=. k = l 3.3. (3.3) k l B A l k â i, ˆb i A, B i { â i (i l) ˆbi = â k (i = l) B = (b i j) k = l n j= b lj b lj = B. ˆb k = ˆb l B = 0 b lj = a kj, b lj = ã lj n j= a kjã lj = 0. t A A = (a ij ) i,j n M nn (K) { n A (k = l) a ik ã il = 0 (k l) i= k = l A A = (a ij ) i,j n M nn (K) à à = (ã ij ) i,j n t ÃA = A t à = A I n. A A = t Ã. A 46

48 K = R or C x, y K A n M nn (K) x + y x 0 0 y x + y x.... A n =. 0 y x + y x 0 0 y x + y A n = x n + x n y xy n + y n. A = x + y, A 2 = x 2 + xy + y 2 n =, 2 n 3,..., n y x x + y x... A n = (x + y) A n x 0 y x + y x 0 0 y x + y = (x + y) A n xy A n 2 n, n 2 A n = (x + y)(x n + x n 2 y y n ) xy(x n 2 + x n 3 y y n 2 ) = x n + x n y xy n + y n 3.7. A M mm (K), B M mn (K), C M nn (K) A B 0 C = A C (3.4) [ n C m ] 47

49 3.8. A, B M nn (K) A B B A = A + B A B [ i =,..., n n + i n j =,..., n j n + j ] 48

50 3.4 () (2) (x + y 2)(x + y + 2)(x y) 2 (3) 8abcd 3.5 ( x 4 ) 3 x 4 = (x+)(x )(x 2 +) = 0. x x =,. 49

51 Chapter e, e 2, e 3 R 2 e = 0, e 2 =, e 3 = 0 F : 0 0 R 3 R 3 (e, e 2, e 3 ) A = linear map f =, f 2 =, f 3 = 0 0 Af = f, Af 2 = f 2, Af 3 = 2f 3 rank (f, f 2, f 3 ) = 3 (f, f 2, f 3 ) R 3 F (f, f 2, f 3 )

52 x x = x 2 R 3 (f, f 2, f 3 ) y 2 x 3 y 3 x y x = x 2 = y f + y 2 f 2 + y 3 f 3 = 0 y 2 x 3 0 y 3 P = 0, y = 0 x = P y y y 2 y 3 y 0 P = 0 4. K = R or C, V K-vector space (e,, e n ), (f,, f n ) V v V (e,, e n ) x =. (f,, f n ) y =. y y n x x n K n, K n x y p j f j (e,, e n ). p nj f j = n i= p ije i v = y f + + y n f n (e,, e n ) 5

53 x x n p p n. = y. + + y n p n. p nn p p n y =... p n p nn y n (4.) P = (p ij ) i,j n M nn (K) x = P y. P (e,, e n ) (f,, f n ) 4... V K-vector space, (e,, e n ), (f,, f n ) V f j (e,, e n ) p j K n P M nn (K) P = (p p n ) P j p j P v V v (e,, e n ) x K n, (f,, f n ) y K n x = P y. P (f,, f n ) (e,, e n ) Q M nn (K) v V x, y K n v (e,, e n ), (f,, f n ) x = P y, y = Qx. x = P Qx, y = QP y. x, y K n P Q = QP = I n. P Q = P U K-vector space, (e,, e n ) U P M nn (K) U (f,, f n ) (e,, e n ) (f,, f n ) P. P = (p ij ) j =,, n f j = p j e + + p nj e j rank (f,, f n ) = rank P = n (f,, f n ).3. (f,, f n ) U 52

54 f,, f n (e,, e n ) (f,, f n ) P (f,, f n ) V = T 2 (R) V (, x, x 2 ) ( + x, 2x, + x + x 2 ) + x, 2x, + x + x 2 (, x, x 2 ), 2, (, x, x 2 ) ( x, 2x, + x + x 2 ) U, V K-vector space, f L(U, V ) (e,, e n ), (f,, f n ) : U, (a,, a m ), (b,, b m ) : V, P M nn (K) : (e,, e n ) (f,, f n ), Q M mm (K) : (a,, a m ) (b,, b m ), A M mn (K) : (e,, e n ), (a,, a m ) f, B M mn (K) : (f,, f n ), (b,, b m ) f A B U (f,,f n) f K n B V (b,,b n ) K m P K n (e,,e n) A U f Q K m (a,,a n ) V u U v = f(u) V u (e,, e n ), (f,, f n ) x, y K n v (a,, a n ), (b,, b n ) z, w K m z = Ax, w = By, x = P y, z = Qw. 53

55 Qw = z = Ax = AP y w = Q AP y. y K n B = Q AP U, V, f B = Q AP U = V, (a,, a m ) = (e,, e n ), (b,, b m ) = (f,, f n ) B = P AP A, B M nn (K) P M nn (K) B = P AP A B A B B = P AP = P A P = A U K-vector space, f L(U), (e,, e n ) U A (e,, e n ) f B A U (f,, f n ) (f,, f n ) f B e = 0, e 2 =, e 3 = 0, f =, f 2 = 0 0 2, f 3 = 2 () R 3 (e, e 2, e 3 ) (f, f 2, f 3 ) 2 0 (2) T L(R 3, R 3 ) (e, e 2, e 3 ) T (f, f 2, f 3 ) 54

56 . () P P = 2 2 (2) T (f, f 2, f 3 ) = = K = R or C () U K-vector space, f L(U) λ K, u U (u 0) f(u) = λu λ f (eigenvalue) u f λ (eigenvector) (2) A M nn (K) λ K, x K n (x 0) Ax = λx λ A x A λ K-vector space U, f L(U), (e,, e n ) U A M nn (K) f (e,, e n ) u U (e,, e n ) x K n λ K f(u) = λu Ax = λx 55

57 () A n n λ,, λ n K λ λ A = λ n A λ,, λ n. e,, e n K n e = 0.,, e n = 0. 0 λ i e i 0 (2) U = C (R) = {f f : R R, f R } d : dx C (R) C (R) f C (R) d u = du dx dx d f dx C (R) d dx λ R df dx = λf C R f(x) = Ceλx d dx A M nn (K) λ K F A (λ) = A λi n λ K A F A (λ) = 0 A M nn (K) F A (λ) λ n F A (λ) A characteristic polynomial B = P AP. x K n, x 0 Ax = λx ker (A λi n ) {0} A λi n = U K-vector space, f L(U) (e,, e n ), (f,, f n ) U A, B f (e,, e n ), (f,, f n ) F A (λ) = F B (λ). f f A F A (λ) F f (λ) f 56

58 A B P M nn (K) B = P AP. B λi n = P (A λi n )P = A λi n A = 0 3 M 33 (R) 0 3. A λ 2 3 F A (λ) = A λi 3 = 0 λ 3 = ( λ)(λ 4)(λ + 2) 0 3 λ, 4, 2. x y z x x 0 (A I 3 ) y = y = 0 z z 0 x y = z = 0 y = t 0 t R, t 0) z 0 x 4 y z x x 0 (A 4I 3 ) y = y = 0 z z 0 x y = 3x = z. y = t 3 t R, t 0 z 3 x 2, 4 y = z 57

59 5/3 t t R, t 0. ( ) K = R K = C 0 J = F 0 J (λ) = λ 2 + J M 22 (R) F J (λ) = 0 R J J M 22 (C) F J (λ) = 0 λ =, J U K-vector space, f L(U) f (semisimple) U (e,, e n ) λ,, λ n K i =,, n f(e i ) = λ i e i U K-vector space, f L(U) () f (2) U f (3) U f (4) U f. () (2) (2) (4): (e,, e n ) U (e,, e n ) f A U (a,, a n ) f B (e,, e n ) (a,, a n ) P 4..4 B = P AP B (4) (3) (3) (2): U K-vector space, f L(U) λ K f λ E λ (f) E λ (f) = {u u U, f(u) = λu} E λ (f) U subspace λ U E λ (f) {0}. E λ (f) E λ 58

60 U K-vector space, f L(U) f {λ,, λ m } i j λ i λ j m f dim E λi (f) = dim U i= U K-vector space, f L(U) f {λ,, λ m } i j λ i λ j i =,, k u i E λi (f) u + + u k = 0 i =,, k u i = 0.. k k = k i =,, k u i E λi, u + + u k = 0 f(u + + u k ) = f(u ) + + f(u k ) = λ u + + λ k u k = 0 λ u λ k u k + λ k u k = 0 λ k u λ k u k + λ k u k = 0 (λ λ k )u + (λ 2 λ k )u (λ k λ k )u k = 0 i =,, k (λ i λ k )u i = 0. i =,, k λ i λ k u i = 0. u + + u k = u k = : : i =,, m (e i,,, e i,ni ) E λi dim E λi = n i n + + n m = dim U e = (e,,, e,n, e 2,,, e 2,n2,, e m,,, e m,nm ) a i,j K m n i a i,j e i,j = 0 i= j= 59

61 u i = n i j= a i,je i,j m i= u i = k = m i =,, m u i = n i j= a i,je i,j = 0. (e i,,, e i,ni ) j =,, n i a i,j = 0. e n + + n m = dim U.3. e U f U K-vector space, f L(U) f dim U f f dim U f dim U K K = R () (2) λ 0. () 4 3 λ λ (λ 3)., 3. x 2 0 x 0 E : y E y = 0 y = 2x z z 0 { } x = z. E = t 2 t R dim E =. { 0 } E 3 : E E 3 = t 0 t R. dim E 3 =. dime + dim E 3 = 2 < 3 2 λ 2 (2) λ λ (λ 2)., 2. x 2 x 0 E : y E 2 y = 0. x = z z 0 60

62 { 2 } 2y + z. E = y + z 0 y, z R dim U = 2. 0 x 0 2 x 0 E 2 : y E 2 3 y = 0. 2y+z = z 2 4 z 0 { } 0 x = 3y + z. E 2 = t t R 2 dim E 2 =. dim E + dim E 2 = () 4 2 (2) A M nn (K) m N A m = A M nn (K) F A (λ) = Ft A(λ) A t A 4.4. A M nn (K) F A (A) = U K-vector space, f, g L(U) f g g f 6

63 Chapter 5 5. U vector space u U u, v U u, v 5... K = R or C. U K-vector space u, v U (u, v) K U U K (, ) U (innder product) (IP), (IP2), (IP3) (IP): u, v U (v, u) = (u, v). z z K = R (u, v) = (v, u). (IP2): α, β K u, v, w U (αu + βv, w) = α(u, w) + β(v, w). (IP3): u U (u, u) (u, u) = 0 u = 0. (, ) U (U, (, )). (IP) (IP2) α, β K, u, v, w U (w, αu + βv) = α(w, u) + β(w, v). K = R α, β R α = α, β = β. u () u =. u n, v v. v n C n (u, v) = n i= u iv i (, ) C n C n 62

64 x (2) x =. x n, y = y. y n R n (x, y) = n i= x iy i (, ) R n ( ) ( ) R n x y (3) x =, y = R 2 x 2 y 2 (x, y) = 2x y + x 2 y 2 x 2 y x y 2 ( ) ( ) 2 y = (x x 2 ) y 2 ( ) 2 = t x y (, ) R 2 (4) U = C([0, ], C) = {f f : [0, ] C, f [0, ] } U C-vector space f, g U (f, g) = (, ) U 0 f(x)g(x)dx U K-vector space, (, ) U () u U u (, ) u u = (u, u) (2) u, v U (u, v) = 0 u v (, ) U K-vector space (, ) U () e,, e n U (e,, e{ n ) (U, (, )) (orthonormal system) (e i, e j ) = i, j {,, n} (i = j) 0 (i j) (2) e,, e n U (e,, e n ) (U, (, )) (orthonormal base) (e,, e n ) U 63

65 5..3 (e,, e n ) (U, (, )) (e,, e n ) U () R n, C n 5..2-(), (2) i = x i { if i = j,, n e i =., x ij = 0 if i j (e,, e n ) x in ( ) ( ) 0 (2) R (3) (, ) (3) 5..2-(4) N ( 2 sin nπx) n=,2,,n U K-vector space, (, ) U U. (e,, e k ) α e + + α k e k = 0 e i α i = 0. (a,..., a n ) U k =,..., n (e,..., e k ) e,..., e k = a,..., a k k = b = a, e = b / b u = (u, u). ) a = e k = k k < n k b k+ = a k+ (a k+, e i )e i i= b k+ = 0 a k+ = k i= (a k+, e i )e i. e,, e k a,, a k a k+ a,, a k (a,, a k, a k+ ) b k+ 0. e k+ = b k+ / b k+ e k+ = i =,, k n (e k+, e i ) = b k+ ((a k+, e i ) (a k+, e j )(e j, e i )) = j=

66 (e,, e k, e k+ ) a k+ e,, e k, e k+ a,, a k, a k+ e,, e k, e k+. (a,, a n ) (e,, e n ) b = a, e = b b k b k+ = a k+ (a k+ e i )e i, e k+ = b k+ b k+. i= Schmidt ( ) ( ) () C 2 u v u =, v = u 2 v 2 ( ) ( ) i v (u, v) = (u u 2 ) i 2 (, ( )) ( ) 0 (2) a =, a 0 2 = Schmidt C 2 (, ) ( ) u. () (IP), (IP2) u = u = x + iy, u 2 = x 2 + iy 2 x, y, x 2, y 2 R (u, u) = u u + iu u 2 iu 2 u + 2u 2 u 2 = (x + y 2 ) 2 + (x 2 y ) 2 + x y 2 2 (IP3) ( ) i (2) a = e = a. b 2 = a 2 (a, e )e = a 2 ( i)e =. ( ) i ( ( ) ( ) i ) b 2 = e 2 =.,. 0 R n C n 65 v 2 u 2

67 5..8. U K-vector space, (, ) U (e,, e n ) (U, (, )) u, v U (e,, e n ) x x n y.,. (u, v) = n i= x iy i = t xy.. y n (u, v) = (x e + + x n e n, y e + + y n e n ) = n (x i e i, y j e j ) = x i y i i,j=,,n i= 5.. a, b, c, d x = ( x x 2 ), y = ( y ( ) ( ) a b y (x, y) = (x x 2 ) c d y 2 y 2 ) R 2 (, ) R 2 b = c, a > 0 b 2 ad < f, g T 2 (R) (f, g) = f(x)g(x)dx 0 () (, ) T 2 (R) (2) T 2 (R) (, x, x 2 ) Schmidt T 2 (R) a, a 2, a 3 R 3 a = 0, a 2 =, a 3 = Schmidt (a, a 2, a 3 ) R U K-vector space, (, ) U e,, e m U (e,, e m ) u U n i= (e i, u) 2 = u 2 66

68 5.2 U K-vector space, (, ) U U U f f L(U) U (, ) u, v U (f(u), f(v)) = (u, v) f U K-vector space, (, ) U f L(U) (), (2) U dim U = n () (3), (4), (5) () f U (, ) (2) f U u U u = f(u). (3) (e,, e n ) U (f(e ),, f(e n )) (4) (e,, e n ) U P (e,, e n ) f t P P = I n. (5) (e,, e n ) U P (e,, e n ) f p j K n P j (p,, p n ) K n (x, y) = t xy. K n (, ) K n. () (2) (2) (): (u + v, u + v) = (f(u + v), f(u + v)) (u, u) + (v, v) + (u, v) + (v, u) = (f(u), f(u)) + (f(v), f(v)) + (f(u), f(v)) + (f(v), f(u)). u = f(u), v = f(v) (u, v) + (v, u) = (f(u), f(v)) + (f(v), f(u)). (5.) K = R (u, v) = (f(u), f(v)). K = C (5.) v iv i(u, v) + i(v, u) = i(f(u), f(v)) + i(f(v), f(u)). (u, v) (v, u) = (f(u), f(v)) (f(u), f(v)). (5.) (u, v) = (f(u), f(v)). () (3): (f(e i ), f(e j )) = (e i, e j ) (3) (5): 2..4 f(e j ) (e,, e n ) p j 5..8 (f(e i ), f(e j )) = (p i, p j ) K n (p,, p n ) K n 67

69 t p (5) (4): t P =. t p n, P = (p p n ) t P P (i, j) t p i p j = (p i, p j ) K n. (p,, p n ) K n t P P = I n. t P P = t P P = I n = I n. (4) (): u, v (e,, e n ) x, y K n (f(u), f(v)) = (P x, P y) K n = t x t P P y = t xy = (x, y) K n = (u, v) () U K-vector space, (, ) U, f L(U) K = C f (, )) f (U, (, )) unitary transformation K = R f (, ) f (U, (, )) orthogonal transformation (2) P M nn (C) t P P = I n P n unitary matrix P M nn (R) t P P = I n P n orthogonal matrix A M nn (K), A i â i â A =. â n () A (2) t A (3) ( t â,, t â n ) K n. () (2): A t AA = I n. A t A = I n. A t A = I n. t A (2) (3):

70 a, b, c, d, e, f a b c d 2 e f. a = b = c = e = 0. 2, 3 a 2, a 3 a 2 = d 2 + f 2 =. θ R d = 0 cos θ, e = sin θ a 3 = cos π/3 (a2, a 3 ) R 3 = 0 sin π/3 ( ) ( ) d 3/2 θ = π/3 + π/2, π/3 π/2 = 5π/6, π/6. = ±. f / b, b 2, b i 6 b 3 i 6 b 2 i b 3. (a a 2 a 3 ) = â â 2 â 3 tâ 3 = b 3 = 0. (a, a 3 ) = (a 2, a 3 ) = 0 b + b 2 = 0. b 2 + b 2 2 =. b = b 2 = / 2, b 2 = b. b θ [0, 2π) b = (cos θ + i sin θ)/ 2, b 2 = (cos θ + i sin θ)/ U K-vector space, (, ) U (e,, e n ) U U (f,, f n ) (e,, e n ) (f,, f n ). (e,, e n ) (f,, f n ) P P j p j K n p j f j (e,, e n ) (f,, f n ) U (p,, p n ) K n P 69

71 U K-vector space, (, ) U (e,, e n ) (U, (, )) f L(U), A f (e,, e n ) λ,, λ n K () (g,, g n ) i =,, n f(g i ) = λ i g i. (2) P λ 0 0. t 0 λ P AP = λ n (5.2). () (2): 4..4 (e,, e n ) (g,, g n ) P (g,, g n ) f P AP. P AP (5.2) P P = t P. (5.2) (2) (): 4..2 U (g,, g n ) (e,, e n ) (g,, g n ) P P (g,, g n ) t P AP = P AP (g,, g n ) f (5.2) i =,, n f(g i ) = λ i g i A ( M 22 (R) ) ( ) θ R cos θ sin θ cos θ sin θ A = A = sin θ cos θ sin θ cos θ 5.6. () A A (2) A A (3) A, B AB (4) A, B A + B 5.7. () P det P P det P = ± (2) P P P P ± 70

72 5.8. a, b, c a 2i b i c 2i 5.9. A, B M nn (R) det A = det B det (A + B) = 0 ( ) A B 5.0. A, B M nn (R) A+iB B A U K-vector space, (, ) U f L(U) u, v U (f(u), v) = (v, f(u)) K = C f (U, (, )) (Hermite transformation) K = R f (U, (, )) (symmetric transformation) U K-vector space, (, ) U f L(U), (e,, e n ) U A (e,, e n ) f () f (2) t A = A. (3) P λ,, λ n R λ 0 0. t 0 λ P AP = λ n (4) U (g,, g n ) λ,, λ n R i =,, n f(g i ) = λ i g i 7

73 () (2) u, v U (e,, e n ) x, y K n (f(u), v) = (Ax, y) K n = t x t Ay. (u, f(v)) = t xay. () t A = A t A = A U K-vector space, (, ) U f L(U) (U, (, )) K = R f ( ). n N α 0,..., α n C z P (z) = z n + α n z n a z + a 0 λ,..., λ n C P (z) = (z λ )(z λ 2 ) (z λ n ) U (e,..., e n ) f (e,..., e n ) A K = R A () (2) t A = A K = R, C A : C n C n A F A (z) λ C F A (λ) = 0. λ A : C n C n λ u C n, u 0 (, ) n C n λ(u, u) n = (λu, u) n = (Au, u) n = (u, Au) n = (u, λu) n = λ(u, u) n (u, u) 0 λ = λ. λ R. K = C K = R λ R F A (λ) = 0 A λi n A λi n v R, v 0 Av = λv 72

74 (3) (4): (3) (2): (3) D A = P D t P. t A = P D t P. λ i R D = D. t A = A. () (4): n = dim U n = U = C λ C z C f(z) = λz. z, z 2 C (f(z ), z 2 ) = λz z 2, (z 2, f(z 2 )) = λz z 2. () λ = λ λ R. n dim U = n f λ R g λ f g =.3.2 Smidt a,, a n U (g, a,, a n ) U V = a,, a n V = {v v U, (e, v) = 0}. v V (e, f(v)) = (f(e ), v) = λ (e, v) = 0. f(v) V. f V f V V V f V L(V ) V dim V = n V (g 2,, g n+ ) λ 2,, λ n+ R f(g i ) = λ i g i i = 2,, n + (g, g 2,, g n ) U n + (4) A M nn (C) t A = A A M nn (R) t A = A = = f f A A (), (2), (3) () A (2) A p,, p n K n, λ,, λ n R (p,, p n ) K n i =,, n f(p i ) = λ i p i 73

75 (3) P M nn (K) P = (p p n ) P D (3) P AP = t P AP = D. AP = (Ap Ap n ) = (λ p λ n p n ) = P D i 0 0 i 0. λ 0 i 0 λ i λ = λ(λ2 2) = 0 λ = 0, ± 2. 0 α i, 2 α i 0, 2i 2 α i. 2i p = 2 i, p 2 = i 2 0, p 2 = i 2. 2i 2i i P = 2 i i i 2i () A m A m 74

76 (2) A m A m = 0 A = 0 (2) A, B AB (3) A, B A + B (4) A, B AB + BA (5) A, B i(ab BA) 5.2. a a a a 2a i 0 () a a a (2) i 4a i a a a 0 i 2a 5.3. A M 33 (R) 3, 2, A = 3, A = () A (2) A 5.4. A,, A m M nn (C) m i= A i 2 = 0 i =,, m A i = A M nn (C) () I n + ia (2) P = (I n ia)(i n + ia) P P + E 5.6. () U C-vector space, f L(U) u U λ R f 3 (u) = λu (f 3 = f f f) f(u) = λ /3 u 75

77 (2) P, Q P 3 = Q 3 P = Q (3) A A = P x,, x n n n a ij x i x j i,j= A M nn (R) A = (a ij ), x =. x x n R n 2 t xax A M nn (R), x R n A[x] = t xax A[x] A quadratic form) x i x j a ij + a ji (a ij + a ji )/2 a ij A P M nn (R) y y =. y n A[P y] = n 2 λ i y i i= R n A[P y] = t P AP t P AP P A P P AP = t P AP 76

78 5.4.. A M nn (R) p = dim E λ (A), q = λ:λ>0 λ A (p, q) A λ:λ<0 λ A P A λ 0 0. t 0 λ P AP = λ n dim E λ (A) p = λ i > 0 i, q = λ i < 0 i A M nn (R), A[x] A (p, q) A Q M nn (R) A[Qy] = y y p 2 (y p y p+q 2 ) (5.3) R A[Ry] = y y r 2 (y r y r+s 2 ) (5.4) r = p, s = q. P A λ 0 0. t 0 λ P AP = λ n λ,, λ p λ p+,, λ p+q λ p+q+,, λ n 0 S = (s ij ) M nn (R) i =,, p s ii = / λ i, i = p +,, p + q s ii = / λ i, i = p + q +,, n s ii = P S = Q I p 0 0 t QAQ = t S t P AP S = 0 I q

79 I p, I q p, q (5.3) (5.4) I r 0 0 t RAR = 0 I s p + q = rank t QAQ = rank A = rank t RAR = r + s { y } V =. R n i p +, y i = 0, y n { z } V 2 =. R n i < r + z i = 0 z n x Q(V ) R(V 2 ) y V, z V 2 x = Qy = Rz. (5.3), (5.4) A[x] = A[Qy] = y y p 2 0 A[x] = A[Rz] = (z r z r+s 2 ) 0. A[x] = 0 y = = y p = 0. y = 0 x = Ay = 0. Q(V ) R(V 2 ) = {0} n dim (Q(V ) + R(V 2 )) = dim Q(V ) + dim R(V 2 ) = p + (n r) r p. Q R r p r = p. p + q = r + s q = s A M nn (R) (p, q) A[x] y y p 2 (y p y p+q 2 ) (x + x 2 ) 2 + x 3 x A =

80 A ±, 0, 2. A (2, ) λ = 2 0, λ = 2 0, λ = 2 0, λ = A 0 P P = A[P y] = 2y y y Q = P A[Qz] = z 2 + z 2 2 z 3 2. A[x] A A[x] (positive definite) x R n, x 0 A[x] > 0 (positive semi-definite) (non-negative definite) x R n A[x] 0 (negative definite) x R n, x 0 A[x] < 0 (negative semi-definite) (non-positive definite) x R n A[x] A M nn (K) () A[x] A A 79

81 (n, 0) (0, n) (2) A[x] A 0 m n A (m, 0) (0, m) A M nn (C) z =. C n A[z] = n a ij z i z j = t zaz i,j= A[z] A A A[z] A[z] = n a ij z i z j = i,j= z z n n a ji z j z i = A[z] i,j= z C n A[z] R A M nn (C) A[z] P M nn (C) λ,, λ n R w w =. w n C n A[P w] = λ w w + + λ n w n w n (p, q) A M nn (C), A[z] A (p, q) A Q M nn (R) A[Qw] = w w + + w p w p (w p+ w p+ + + w p+q w p+q ) 80

82 R A[Rw] = w w + + w r w r (w r+ w r+ + + w r+s w r+s ) r = p, s = q A A[z] 5.7. () x x 2 + x 2 x (2) x x x 3 + x 3 x (3) 2(x x 2 + x x 3 + x x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 ) 5.8. a 2a i 0 A = i 4a i 0 i 2a A a 5.9. A (a), (b), (c) (a) A (b) P A = t P P (c) B A = B 2 A (c) B A M nn (C) () Q M nn (C) A t A = Q 2 (2) Q () P = Q A P (3) Q () P = Q A L U A = LU L = Q, U = P 5.2. A, B A, B AB 8

83 U, V K-vector space (, ) U, (, ) V U, V f L(U, V ) u U, v V (f(u), v) V = (u, f (v)) U f L(V, U) (e,, e n ) U (g,, g m ) V A M mn (K) (e,, e n ), (g,, g m ) f f t A. (g,, g m )(e,, e n ) t A f L(V, U) u U, v V, x K n u (e,, e n ) y K n v (g,, g m ) (f(u), v) V = (Ax, y) K m = t x t Ay = t x( t Ay) = (x, t Ay) K n = (u, f (v)) U. g L(V, U) u U, v V (f(u), v) V = (u, g(v)) U (u, g(v) f (v)) U = 0. u U g(v) = f (v). g = f () U, V K-vector space (, ) U, (, ) V U, V f L(U, V ) 5.5. f L(V, U) f (adjoint transformation) (2) A M nm (K) t A M mn (K) A U K-vector space, (, ) U f L(U) () f f f U (2) f f = f () U K-vector space, (, ) K f L(U) f f = f f f (normal transformation) (2) A M nn (C) t AA = A t A A (normal matrix) 82

84 U C-vector space, (, ) U f L(U) (e,, e n ) (U, (, ) A M nn (C) (e,, e n ) f () f (2) A (3) P λ,, λ n C λ 0 0. t 0 λ P AP = λ n (4) (U, (, )) (g,, g n ) λ,, λ n C i =,, n f(g i ) = λ i g i.. C R () (2) (3) (4) (4) (): (3) D f (g,, g n ) D f (g,, g n ) D. D D DD = DD. f f = f f. () (4): n = dim U n = λ C f(u) = λu u U n dim U = n + λ f g U g = g E λ (f) h,, h n U (g, h,, h n ) U B (g, h,, h n ) f ( ) ( ) λ ˆb B =, t λ 0 B = 0 C ˆb = (b b n ) M n (C), C M nn (C) f t BB = B t B ( t BB ) B t B (, )- ˆb = 0 λ 0 B = V = h 0 C,, h n v V f(v) V. f V f V ṱ b t C 83

85 V V f V (h,, h n ) C t BB = B t B t CC = C t C. f V V dim V = n V (g 2,, g n+ ) λ 2,, λ n+ i = 2,, n + f(g i ) = λ i g i. (g, g 2,, g n+ ) n + (4) C R A M nn (R) (2) (3) U C-vector space, (, ) U f L(U) u U f(u) = f (u) A M nn (C) A A A B, C A = B + ic A BC = CB Ω {A A M nn (C), t AA = A t A} A, B Ω AB = BA P A Ω t P AP 84

i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................

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